 where (REF ) follows from [1]} with \(n\)  being replaced by \(nL\)  by the fact ().

Now, we provide a relationship between the aggregated pseudo spectral gap \(\gamma _{\rm {aps}}\)  of the sequence \(X_1,X_2,\cdots ,X_n\)  to its mixing time, called aggregated mixing time. First, we introduce the aggregated mixing time of a sequence of random variables.
